Simple Harmonic Oscillations (Copy)

Key Terminologies and Basic Concepts

• Displacement (x):
  • The distance of the oscillating particle from its equilibrium (central/rest) position at any instant
  • It can be positive or negative depending on the direction of motion from the equilibrium
  • Measured in metres (m)
• Amplitude (x₀):
  • The maximum displacement from the equilibrium position
  • Always a positive scalar quantity
  • Represents the maximum extent of oscillation
  • Measured in metres (m)
• Period (T):
  • The time taken for one complete oscillation
  • Measured in seconds (s)
  • Related to frequency by the formula:
    • T = 1/f
• Frequency (f):
  • The number of oscillations per second
  • Measured in hertz (Hz), where 1 Hz = 1 oscillation per second
  • Formula: f = 1/T
• Angular frequency (ω):
  • Also called angular speed in SHM
  • Defines how quickly an oscillation completes its cycle in terms of angle (radians)
  • Formula: ω = 2πf = 2π/T
  • Measured in radians per second (rad/s)
• Phase:
  • Describes the position within the cycle at any moment, measured in radians
  • It represents how far along its oscillation a particle is at a given time
• Phase difference (Δφ):
  • The angular difference in phase between two oscillating particles
  • Measured in radians or degrees
  • One complete cycle corresponds to 2π radians or 360°

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Defining Simple Harmonic Motion (SHM)

• SHM is a type of periodic motion
• Occurs when acceleration is directly proportional to displacement, but acts in the opposite direction
• Mathematical condition for SHM:
  • a ∝ −x
  • Or: a = −ω²x
    • a = acceleration
    • x = displacement
    • ω = angular frequency
• This negative sign indicates that the acceleration is directed towards the equilibrium position
• This restoring force (or acceleration) causes the oscillatory behavior

Equation of Motion for SHM

• The displacement-time equation (general solution):
  • x = x₀ sin(ωt) or x = x₀ cos(ωt)
    • Depends on starting conditions (sine if starting from zero displacement, cosine if starting from max displacement)
• Velocity is the derivative of displacement with respect to time:
  • v = dx/dt
  • If x = x₀ sin(ωt), then v = x₀ω cos(ωt)
• Therefore:
  • v = ±ω√(x₀² − x²)
    • At maximum displacement (x = x₀), v = 0
    • At equilibrium (x = 0), v = ωx₀ (maximum)
• Acceleration is the derivative of velocity:
  • a = dv/dt = −ω²x
  • Again confirming that acceleration is proportional to –x (restoring force model)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Summary of Key SHM Equations

Graphical Representations of SHM

• Displacement-Time Graph:
  • Sinusoidal wave
  • Amplitude = x₀
  • Period = T
  • Zero displacement at multiples of T/2
• Velocity-Time Graph:
  • Also sinusoidal, π/2 radians (90°) out of phase with displacement graph
  • Velocity is maximum at equilibrium (x = 0)
• Acceleration-Time Graph:
  • Also sinusoidal, π radians (180°) out of phase with displacement graph
  • Maximum acceleration at maximum displacement

Phase Relationships

• These phase differences explain the motion visually and are key in interpreting SHM graphs

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Important Graph Features

1. Displacement-Time (x vs. t)

• Shape: sine or cosine wave
• Zero crossings at: t = 0, T/2, T, 3T/2…
• Peaks at: t = T/4, 3T/4, etc.

2. Velocity-Time (v vs. t)

• Shape: cosine or negative sine
• Zero velocity at displacement maximum
• Peak velocity at equilibrium

3. Acceleration-Time (a vs. t)

• Shape: sine or cosine wave (inverted from displacement)
• Acceleration is maximum at max displacement
• Acceleration is zero at equilibrium

Energy in SHM (Linked Concept)

• Total mechanical energy (E) in SHM is constant if no damping
• Composed of:
  • Kinetic energy (KE) = ½mv²
  • Potential energy (PE) = ½mω²x²
• Total Energy (E) = ½mω²x₀²

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course

Conditions for SHM

For a system to undergo SHM:

• Must have a restoring force directed towards the equilibrium
• Restoring force must be directly proportional to displacement:
  • F = −kx (Hooke’s law for springs)
• No energy lost to friction or damping (in ideal SHM)
• Oscillations should occur about a fixed equilibrium point

Examples of SHM Systems

• Mass-spring system (horizontal or vertical):
  • Obeys F = −kx, so SHM motion results
• Simple pendulum (for small angles):
  • Approximate SHM where restoring force = −mg sin θ ≈ −mgθ
  • Valid only for θ < 10°

Real-Life Applications of SHM

• Shock absorbers in cars (damped SHM)
• Oscillating springs in lab experiments
• Seismometers for earthquake detection
• MEMS sensors in mobile phones (gyroscopes/accelerometers)
• Musical instruments (vibrating strings, air columns)

Written and Compiled By Sir Hunain Zia, World Record Holder With 154 Total A Grades, 7 Distinctions and 11 World Records For Educate A Change A2 Level Physics Full Scale Course